{"id":626464,"date":"2023-09-12T02:31:06","date_gmt":"2023-09-12T02:31:06","guid":{"rendered":"https:\/\/www.indcareer.com\/schools\/?p=626464"},"modified":"2023-09-12T02:31:13","modified_gmt":"2023-09-12T02:31:13","slug":"kc-sinha-exercise-19-1-mathematics-solution-class-12-chapter-19-indefinite-integrals","status":"publish","type":"post","link":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-19-1-mathematics-solution-class-12-chapter-19-indefinite-integrals\/","title":{"rendered":"KC Sinha: Exercise 19.1- Mathematics Solution Class 12 Chapter 19 Indefinite Integrals"},"content":{"rendered":"\n

<\/span><\/p>\n\n\n\n

Question 1<\/strong>
\u222b(x2<\/sup>-5x+7)dx
Sol :
= \u222bx2<\/sup>dx-5\u222bxdx+7\u222bdx<\/p>\n\n\n\n

[\u2235 \u222bxn<\/sup>dx=$\\dfrac{x^{n+1}}{n+1}$]<\/p>\n\n\n\n

=$\\dfrac{x^3}{3}-\\dfrac{5x^2}{2}+7x+c$<\/p>\n\n\n\n

Question 2<\/strong><\/p>\n\n\n\n

\u222b(ax3<\/sup>-bx2<\/sup>+cx+d)dx
Sol :<\/p>\n\n\n\n

=a\u222bx3<\/sup>dx+b\u222bx2<\/sup>dx+c\u222bxdx+d\u222bdx<\/p>\n\n\n\n

=$a\\dfrac{x^4}{4}+b\\dfrac{x^3}{3}+c\\dfrac{x^2}{2}+dx+c$<\/p>\n\n\n\n

Question 3<\/strong><\/p>\n\n\n\n

$ \\displaystyle\\int \\left(x+\\dfrac{1}{x}+2\\right)$
Sol :
= \u222bxdx+$\\int \\dfrac{1}{x}dx$+2\u222bdx<\/p>\n\n\n\n

=$\\dfrac{x^2}{2}$+log|x|+2x+c<\/p>\n\n\n\n

Question 4<\/strong><\/p>\n\n\n\n


\u222b(x1\/3<\/sup>+2x1\/2<\/sup>+x3\/2<\/sup>)dx
Sol :
=\u222bx1\/3<\/sup>dx+2\u222bx1\/2<\/sup>dx+\u222bx3\/2<\/sup>dx<\/p>\n\n\n\n

=$\\dfrac{x^{\\frac{1}{3}+1}}{\\frac{1}{3}+1}+2 \\times \\dfrac{x^{\\frac{1}{2}+1}}{\\frac{1}{2}+1}+\\dfrac{x^{\\frac{3}{2}+1}}{\\frac{3}{2}+1}$+c<\/p>\n\n\n\n

=$\\dfrac{x^{4\/3}}{4\/3}+\\dfrac{2x^{3\/2}}{3\/2}+\\dfrac{x^{5\/2}}{5\/2}$+c<\/p>\n\n\n\n

=$\\dfrac{3}{4}x^{4\/3}+\\dfrac{2}{3}\\times 2 x^{3\/2}+\\dfrac{2}{5}x^{5\/2}$+c<\/p>\n\n\n\n

=$\\dfrac{3}{4}x^{4\/3}+\\dfrac{4}{3}x^{3\/2}+\\dfrac{2}{5}x^{5\/2}$+c<\/p>\n\n\n\n

Question 5<\/strong><\/p>\n\n\n\n

$\\displaystyle\\int \\dfrac{x^2-2x+3}{x^4}dx$<\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

=$\\displaystyle\\int \\dfrac{x^2}{x^4}dx-2\\displaystyle\\int \\dfrac{x}{x^4}dx+3\\displaystyle\\int \\dfrac{1}{x^4}dx$<\/p>\n\n\n\n

=$\\displaystyle\\int \\dfrac{1}{x^2}dx-2\\displaystyle\\int \\dfrac{1}{x^3}dx+3\\displaystyle\\int \\dfrac{1}{x^4}dx$<\/p>\n\n\n\n

=$\\displaystyle\\int x^{-2}dx-2\\displaystyle\\int x^{-3}dx+3\\displaystyle\\int x^{-4}dx$<\/p>\n\n\n\n

=$\\dfrac{x^{-2+1}}{-2+1}-2\\dfrac{x^{-3+1}}{-3+1}+3\\dfrac{x^{-4+1}}{-4+1}$+c<\/p>\n\n\n\n

=$\\dfrac{x^{-1}}{-1}-2\\dfrac{x^{-2}}{-2}+3\\dfrac{x^{-3}}{-3}$+c<\/p>\n\n\n\n

=$-\\dfrac{1}{x}+\\dfrac{1}{x^2}-\\dfrac{1}{x^3}$+c<\/p>\n\n\n\n

Question 6<\/strong>
$\\displaystyle\\int \\left(\\sqrt{x}+\\dfrac{1}{\\sqrt{x}}^2 \\right)dx$
Sol:
=$\\displaystyle\\int \\left\\{(\\sqrt{x})^2+\\left(\\dfrac{1}{\\sqrt{x}}\\right)+2 .\\sqrt{x}.\\dfrac{1}{\\sqrt{x}}\\}$<\/p>\n\n\n\n

=$\\displaystyle\\int \\left(x+\\dfrac{1}{x}+2\\right)dx$<\/p>\n\n\n\n

=$\\displaystyle\\int xdx+\\displaystyle\\int \\dfrac{1}{x} dx+2\\displaystyle\\int dx$<\/p>\n\n\n\n

=$\\dfrac{x^2}{2}$+log|x|+2x+c<\/p>\n\n\n\n

Question 7<\/strong><\/p>\n\n\n\n

$\\displaystyle\\int \\dfrac{(x-3)^2}{\\sqrt{x}}dx$<\/p>\n\n\n\n

Sol:<\/p>\n\n\n\n

=$\\displaystyle\\int \\dfrac{x^2+9-2.x.3}{\\sqrt{x}}dx$<\/p>\n\n\n\n

=$\\displaystyle\\int \\dfrac{x^2}{\\sqrt{x}}dx+9\\displaystyle\\int \\dfrac{1}{\\sqrt{x}}dx-6\\displaystyle\\int \\dfrac{x}{\\sqrt{x}}dx$<\/p>\n\n\n\n

=$\\displaystyle\\int x^{2-\\frac{1}{2}}dx+9\\displaystyle\\int x^{-1\/2}dx-6\\displaystyle\\int \\sqrt{x}dx<\/p>\n\n\n\n

=$\\displaystyle\\int x^{3\/2}dx+9\\displaystyle\\int x^{-1\/2}dx-6\\displaystyle\\int x^{1\/2}dx<\/p>\n\n\n\n

=$\\dfrac{x^{5\/2}}{5\/2}+9\\dfrac{x^{1\/2}}{1\/2}-6\\dfrac{x^{3\/2}}{3\/2}$<\/p>\n\n\n\n

=$\\dfrac{2}{5}x^{5\/2}+18.x^{1\/2}-4x^{3\/8}+c$<\/p>\n\n\n\n

=$\\dfrac{2}{5}\\sqrt{x}(x^2+45-10x)+c$<\/p>\n\n\n\n

=$\\dfrac{2}{5}\\sqrt{x}(x^2-10x+45)+c$<\/p>\n\n\n\n

Question 8<\/strong><\/p>\n\n\n\n

$\\displaystyle \\int \\dfrac{(x^3+1)(x-2)}{x^2-x-2}dx$<\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

[(a3<\/sup>+b3<\/sup>)=(a+b)(a2<\/sup>+b2<\/sup>-ab)]<\/p>\n\n\n\n

=$\\displaystyle \\int \\dfrac{(x+1)(x^2+1-x)(x-2)}{x^2-x-2}$<\/p>\n\n\n\n

=$\\displaystyle \\int \\dfrac{(x+1)(x-2)(x^2+1-x)}{x^2-2x+x-2}$<\/p>\n\n\n\n

=$\\displaystyle \\int \\dfrac{(x+1)(x-2)(x^2+1-x)}{x(x-2)+1(x-2)}$<\/p>\n\n\n\n

=$\\displaystyle \\int \\dfrac{(x+1)(x-2)(x^2+1-x)}{(x+1)(x-2)}$<\/p>\n\n\n\n

=$\\displaystyle\\int (x^2+1-x)dx$<\/p>\n\n\n\n

=$\\displaystyle\\int x^2dx+1\\displaystyle\\int dx-\\displaystyle\\int xdx$<\/p>\n\n\n\n

=$\\dfrac{x^3}{3}+x-\\dfrac{x^2}{2}$+c<\/p>\n\n\n\n

Question 9<\/strong>
$\\displaystyle\\int (ax^2+bx+c)dx$
Sol :
=$\\displaystyle\\int ax^2dx+\\displaystyle\\int bxdx+\\displaystyle\\int cdx$<\/p>\n\n\n\n

=$a\\displaystyle\\int x^2dx+b\\displaystyle\\int xdx+c\\displaystyle\\int dx$<\/p>\n\n\n\n

=$a.\\dfrac{x^3}{3}+b\\dfrac{x^2}{2}+cx$+k<\/p>\n\n\n\n

Question 10<\/strong><\/p>\n\n\n\n

$\\displaystyle\\int (3x^3+4x^3)dx$
Sol :
=$\\displaystyle\\int 3x^2dx+\\displaystyle\\int 4x^3dx$<\/p>\n\n\n\n

=$3 \\displaystyle\\int x^2dx+4\\displaystyle\\int x^3dx$<\/p>\n\n\n\n

=$3\\dfrac{x^3}{3}+4\\dfrac{x^4}{4}$+c<\/p>\n\n\n\n

=x3<\/sup>+x4<\/sup>+c<\/p>\n\n\n\n

Question 11<\/strong>
$\\displaystyle\\int \\dfrac{x^3+5x^2-4}{x^2}dx$
Sol :
=$\\displaystyle\\int \\dfrac{x^3}{x^2}dx+5\\displaystyle\\int \\dfrac{x^2}{x^2}dx-4\\displaystyle\\int \\dfrac{1}{x^2}dx$<\/p>\n\n\n\n

=$\\displaystyle\\int x dx+5\\displaystyle\\int dx-4\\displaystyle\\int x^{-2}dx$<\/p>\n\n\n\n

=$\\dfrac{x^2}{2}+5x-4 \\times \\dfrac{x^{-2+1}}{-2+1}$+c<\/p>\n\n\n\n

=$\\dfrac{x^2}{2}+5x-4 \\times \\dfrac{x^{-1}}{-1}$+c<\/p>\n\n\n\n

=$\\dfrac{x^2}{2}+5x + \\dfrac{4}{x}$+c<\/p>\n\n\n\n

Question 12<\/strong>
$\\int \\frac{x^{3}-1}{x^{2}} d x$
Sol :
$=\\int \\frac{x^{3}}{x^{2}} d x-\\int \\frac{1}{x^{2}} d x$<\/p>\n\n\n\n

$\\int x d x-\\int x^{-2} d x$<\/p>\n\n\n\n

$=\\frac{x^{2}}{2}-\\frac{x^{-2+1}}{-2+1}+c$<\/p>\n\n\n\n

$\\frac{x^{2}}{2}-\\frac{x^{-1}}{-1}+c$<\/p>\n\n\n\n

$=\\frac{x^{2}}{2}+\\frac{1}{x}+c$<\/p>\n\n\n\n

Question 13<\/strong><\/p>\n\n\n\n

$\\int x^{2}\\left(1-\\frac{1}{x^{2}}\\right) d x$<\/p>\n\n\n\n

Sol:<\/p>\n\n\n\n

Given, $\\int x^{2}\\left(1-\\frac{1}{x^{2}}\\right) d x$<\/p>\n\n\n\n

$\\int x^{2} d x-\\int x^{2}+\\frac{1}{x^{2}} d x$<\/p>\n\n\n\n

$=\\int x^{2} d x-\\int d x$<\/p>\n\n\n\n

$=\\frac{x^{3}}{3}-x+c$<\/p>\n\n\n\n

Question 14<\/strong>
$\\int \\frac{x^{3}+3 x+4}{\\sqrt{x}} d x$
Sol:
$=\\int \\frac{x^{3}}{\\sqrt{x}} d x+3 \\int \\frac{x}{\\sqrt{x}} d x+4 \\int \\frac{1}{\\sqrt{x}} d x$<\/p>\n\n\n\n

$=\\int \\frac{x^{3}}{x^{1 \/ 2}} d x+3 \\int \\frac{\\sqrt{x} \\cdot \\sqrt{x}}{\\sqrt{x}}+4 \\int \\frac{1}{x^{1 \/ 2}} d x$<\/p>\n\n\n\n

$=\\int x^{3-\\frac{1}{2}} d x+3 \\int \\sqrt{x} d x+4 \\int x^{-1 \/ 2} d x$<\/p>\n\n\n\n

$=\\int x^{5 \/ 2} d x+3 \\int x^{1 \/ 2} d x+4 \\int x^{-1 \/ 2} d x$<\/p>\n\n\n\n

$=\\frac{x^{-\\frac{5}{2}+1}}{\\frac{5}{2}+1}+3 \\cdot \\frac{x^{\\frac{1}{2}+1}}{\\frac{1}{2}+1}+4 \\cdot \\frac{x^{-\\frac{1}{2}+1}}{-\\frac{1}{2}+1}$<\/p>\n\n\n\n

=\\frac{2}{7} \\cdot x^{\\frac{7}{2}}+3 \\frac{x^{3 \/ 2}}{\\frac{3}{2}}+\\frac{4 \\cdot \\frac{x^{1\/2}}{\\frac{1}{2}}}{\\frac{1}{2}}<\/p>\n\n\n\n

=$\\dfrac{2}{7}.x^{\\frac{3}{2}}+2x^{3\/2}+8x^{1\/2}+c$<\/p>\n\n\n\n

Question 15<\/strong>
$\\int\\left(x^{2 \/ 3}+1\\right) d x$
Sol :
$=\\int x^{\\frac{2}{3}} d x+\\int 1 d x$<\/p>\n\n\n\n

$=\\frac{x^{\\frac{2}{3}+1}}{\\frac{2}{3}+1}+x+c$<\/p>\n\n\n\n

$=\\frac{x^{\\frac{5}{3}}}{\\frac{5}{3}}+x+c$<\/p>\n\n\n\n

$=\\frac{3}{5} \\cdot x^{\\frac{5}{3}}+x+c$<\/p>\n\n\n\n

Question 16<\/strong><\/p>\n\n\n\n

$\\int(1-x) \\sqrt{x} d x$
Sol :
$=\\int x^{\\frac{1}{2}} d x-\\int x \\cdot x^{\\frac{1}{2}} d x$<\/p>\n\n\n\n

$\\left(x^{m} \\cdot x^{\\pi}=x^{m+n}\\right)$<\/p>\n\n\n\n

$=\\int x^{\\frac{1}{2}} d x-\\int x^{1+\\frac{1}{2}} d x$<\/p>\n\n\n\n

$=\\frac{x^{\\frac{1}{2}+1}}{\\frac{1}{2}+1}-\\frac{x^{1+\\frac{1}{2}+1}}{1+\\frac{1}{2}+1}+c$<\/p>\n\n\n\n

$=\\frac{x^{\\frac{3}{2}}}{\\frac{2}{3}}-\\frac{x^{\\frac{5}{2}}}{\\frac{5}{2}}+c$<\/p>\n\n\n\n

$=\\frac{3}{2} \\cdot x^{3\/2}$<\/p>\n\n\n\n

Question 17<\/strong>
$\\int\\left(\\sqrt{x}-\\frac{1}{\\sqrt{x}}\\right)^{2} d x$
Sol :
$=\\int\\left(x+\\frac{1}{x}-2\\right) d x$<\/p>\n\n\n\n

$=\\int x d x+\\int \\frac{1}{x} d x-2 \\int d x$<\/p>\n\n\n\n

=$\\frac{x^{2}}{2}+\\log |x|-2 x+c$<\/p>\n\n\n\n

Question 18<\/strong>
$\\int \\sqrt{x}\\left(3 x^{2}+2 x+3\\right) d x$
Sol :
$=\\int \\sqrt{x} \\cdot 3 x^{2} d x+\\int \\sqrt{x} \\cdot 2 x d x+\\int \\sqrt{x} \\cdot 3 d x$<\/p>\n\n\n\n

$=3 \\int x^{\\frac{1}{2}} \\cdot x^{2} d x+2 \\int x^{\\frac{1}{2}} \\cdot x d x+3 \\int x^{\\frac{1}{2}} d x$<\/p>\n\n\n\n

$=3 \\int x^{\\frac{1}{2}+2} d x+2 \\int x^{\\frac{1}{2}+1} d x+3 \\int x^{\\frac{1}{2}} d x$<\/p>\n\n\n\n

$=3 \\int x^{\\frac{5}{3}} d x+2 \\int x^{\\frac{3}{2}} d x+3 \\int x^{\\frac{1}{2}} d x$<\/p>\n\n\n\n

$=3 . \\dfrac{x^{\\frac{5}{3}+1}}{\\frac{5}{3}+1}+2. \\dfrac{x^{\\frac{3}{2}+1}}{\\frac{3}{2}+1} +3.\\dfrac{x^{\\frac{1}{2}+1}}{\\frac{1}{2}+1} +c$<\/p>\n\n\n\n

$=3 \\cdot \\frac{x^{\\frac{7}{2}}}{\\frac{7}{2}}+2 \\cdot \\frac{x^{\\frac{5}{2}}}{\\frac{5}{2}}+3 \\frac{x^{\\frac{3}{2}}}{\\frac{3}{2}}$<\/p>\n\n\n\n

$=\\frac{6}{7} \\cdot x^{\\frac{7}{2}}+\\frac{4}{5} x^{\\frac{5}{2}}+2 x^{\\frac{3}{2}}+c$<\/p>\n\n\n\n

Question 19<\/strong><\/p>\n\n\n\n

$\\int \\frac{x^{3}-x^{2}+x-1}{x-1} d x$<\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

$=\\int \\frac{x^{2}(x-1)+(x-1)}{x-1} d x$<\/p>\n\n\n\n

$=\\int \\frac{(x+1)\\left(x^{2}+1\\right)}{x-1} d x$<\/p>\n\n\n\n

$=\\int\\left(x^{2}+1\\right) d x$<\/p>\n\n\n\n

$=\\int x^{2} d x+\\int d x$<\/p>\n\n\n\n

$=\\frac{x^{3}}{3}+x+c$<\/p>\n\n\n\n

Question 20<\/strong>
$\\int\\left(\\sqrt{x}+\\frac{1}{\\sqrt{x}}\\right) d x$
Sol :
$=\\int x^{\\frac{1}{2}} d x+\\int x^{-\\frac{1}{2}} d x$<\/p>\n\n\n\n

$=\\frac{x^{\\frac{1}{2}+1}}{\\frac{1}{2}+1}+\\frac{x^{-\\frac{1}{2}+1}}{-\\frac{1}{2}+1}+c$<\/p>\n\n\n\n

$=\\frac{x^{\\frac{3}{2}}}{\\frac{3}{2}}+\\frac{x^{\\frac{2}{2}}}{\\frac{1}{2}}+c$<\/p>\n\n\n\n

$=\\frac{2}{3} \\cdot x^{\\frac{3}{2}}+2 x^{\\frac{1}{2}}+c$<\/p>\n\n\n\n

$=\\frac{2}{3} x^{\\frac{3}{2}}+2 \\sqrt{x}+c$<\/p>\n\n\n\n

Question 21<\/strong>
$\\int\\left(4 e^{3 x}+1\\right) d x$
Sol :
$=y \\int e^{3 x} d x+\\int d x$<\/p>\n\n\n\n

$=4 \\frac{e^{3 x}}{3}+x+c$<\/p>\n\n\n\n

$=\\frac{4}{3} \\cdot e^{3 x}+x+c$<\/p>\n\n\n\n

Question 22<\/strong>
$\\int\\left(x^{\\frac{3}{2}}+2 e^{x}-\\frac{1}{x}\\right) d x$
Sol :
$=\\int x^{\\frac{3}{2}} d x+2 \\int e^{x}-\\int \\frac{1}{x} d x$<\/p>\n\n\n\n

$=\\frac{x^{\\frac{3}{2}+1}}{\\frac{3}{2}+1}+2 e^{x}-\\log |x|+c$<\/p>\n\n\n\n

=$\\dfrac{2}{5}.x^{\\frac{5}{2}}+2.e^x-log|x|+c$<\/p>\n\n\n\n

Question 23<\/strong><\/p>\n\n\n\n

$\\int\\left(2 x^{2}+e^{x}\\right) d x$<\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

$=2 \\int x^{2} d x+\\int e^{x} d x$<\/p>\n\n\n\n

$=2 \\int x^{2} d x+\\int e^{x} d x$<\/p>\n\n\n\n

$=2 \\cdot \\frac{x^{3}}{3}+e^{x}+c$<\/p>\n\n\n\n

$=\\frac{2}{3} \\cdot x^{3}+e^{x}+c$<\/p>\n\n\n\n

Question 24<\/strong>
$\\int(\\sin x+\\cos x) d x$
Sol :
$=\\int \\sin x d x+\\int \\cos x d x$<\/p>\n\n\n\n

$=-\\cos x+\\sin x+c$<\/p>\n\n\n\n

Question 25<\/strong>
$\\int \\frac{2-3 \\sin x}{\\cos ^{2} x} d x$
Sol :
$=\\int \\frac{2}{\\cos ^{2} x} d x-3 \\int \\frac{\\sin x}{\\cos ^{2} x} d x$<\/p>\n\n\n\n

=2 \\int \\sec ^{2} x d x-3 \\int \\frac{\\sin x}{\\cos x \\cdot \\cos x} d x<\/p>\n\n\n\n

$=2 \\int \\sec ^{2} x d x-3 \\int \\tan x \\cdot \\sec x d x$<\/p>\n\n\n\n

=2 \\tan x-3 \\sec x+c<\/p>\n\n\n\n

Question 26<\/strong>
$\\int \\operatorname{cosec} x(\\operatorname{cosec} x+\\cot x) d x$
Sol :
$=\\int \\operatorname{cosec} x \\cdot \\operatorname{cosec} x d x+\\int \\operatorname{cosec} x \\cdot \\cot x d x$<\/p>\n\n\n\n

$=\\int \\operatorname{cosec}^{2} x d x+\\int \\operatorname{cosec} x \\cdot \\cot x d x$<\/p>\n\n\n\n

$=\\cot x-\\operatorname{cosec} x+c$<\/p>\n\n\n\n

$=(-\\cot x+\\operatorname{cosec} x)+c$<\/p>\n\n\n\n

Question 27<\/strong>
$\\int \\sec x(\\sec x+\\tan x) d x$
Sol :
$=\\int \\sec ^{2} x d x+\\int \\sec x \\cdot \\tan x d x$<\/p>\n\n\n\n

$=\\tan x+s e c x+c$<\/p>\n\n\n\n

Question 28<\/strong>
$\\int \\frac{1-\\sin x}{\\cos ^{2} x} d x$
Sol :
$=\\int \\frac{1}{\\cos^2 x} d x-\\int \\frac{\\sin x}{\\cos x.cosx} d x$<\/p>\n\n\n\n

$=\\int \\sec ^{2} x d x-\\int \\sec x \\cdot \\tan x d x$<\/p>\n\n\n\n

$=\\tan x-\\sec x+c$<\/p>\n\n\n\n

Question 29<\/strong>
$\\int\\left(\\sec ^{2} x+\\operatorname{cosec}^{2} x\\right) d x$
Sol :
$=\\int \\sec ^{2} x d x+\\int \\operatorname{cosec}^{2} x d x$<\/p>\n\n\n\n

$=\\tan x-\\cot x+c$<\/p>\n\n\n\n

Question 30<\/strong>
$\\int\\left(\\sin 2 x-4 e^{3 x}\\right) d x$
Sol :
$\\int \\sin 2 x d x-4 \\int e^{3 x} d x$<\/p>\n\n\n\n

$=\\frac{-\\cos 2 x}{2}-\\frac{4 e^{3 x}}{3}+c$<\/p>\n\n\n\n

$=\\frac{-\\cos 2 x}{2}-\\frac{y}{3} \\cdot e^{3 x}+c$<\/p>\n\n\n\n

Question 31<\/strong><\/p>\n\n\n\n

$\\int\\left(2 x-3 \\cos x+e^{x}\\right) d x$<\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

$=2 \\int x d x-3 \\int \\cos x d x+\\int e^{x} d x$<\/p>\n\n\n\n

$=2\\times \\frac{x^{2}}{2}-3 \\sin x+e^{x}+c$<\/p>\n\n\n\n

$x^{2}-3 \\sin x+e^{x}+c$<\/p>\n\n\n\n

Question 32<\/strong>
$\\int\\left(2 x^{2}-3 \\sin x+5 \\sqrt{x}\\right) d x$
Sol :
$=2 \\int x^{2} d x-3 \\int \\sin x d x+5 \\int \\sqrt{x} d x$<\/p>\n\n\n\n

=$2 \\cdot \\frac{x^{3}}{3}-3(-\\cos x)+5 \\cdot \\frac{x^{-2}+1}{\\frac{1}{2}+1}$+c<\/p>\n\n\n\n

=$\\frac{2}{3} x^{3}+3 \\cos x+5 \\cdot \\frac{x^{3\/2}}{\\frac{3}{2}}+c$<\/p>\n\n\n\n

=$\\frac{2}{3} x^{3}+3 \\cos x+\\frac{10}{3} \\cdot x^{\\frac{3}{2}}+c$<\/p>\n\n\n\n

Question 33<\/strong><\/p>\n\n\n\n

(i)$\\int\\left(5 \\cos x-4 \\sin x+\\dfrac{1}{\\cos ^{2} x}\\right) d x$<\/p>\n\n\n\n

Sol :
$=5 \\int \\cos x d x-4 \\int \\sin x+\\int \\sec ^{2} x d x$<\/p>\n\n\n\n

=5sinx-4(-cosx)+tanx+c<\/p>\n\n\n\n

=5sinx+4cosx+tanx+c<\/p>\n\n\n\n

(ii) $\\int \\frac{\\sin ^{2} x-\\cos ^{2} x}{1-2 \\sin ^{2} x \\cdot \\cos ^{2} x} d x$<\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

$\\int \\frac{\\sin ^{2} x-\\cos ^{2} x}{1-2 \\sin ^{2} x \\cdot \\cos ^{2} x} d x$<\/p>\n\n\n\n

$\\int \\frac{\\left(\\sin ^{4} x\\right)^{2}-\\left(\\cos ^{4} x\\right)^{2}}{1-2 \\sin ^{2} x \\cdot \\cos ^{2} x} d x$<\/p>\n\n\n\n

$\\int \\frac{\\left(\\sin ^{4} x-\\cos ^{4} x\\right)\\left(\\sin ^{4} x+\\cos ^{4} x\\right)}{1-2 \\sin ^{2} x \\cdot \\cos ^{2} x} d x$<\/p>\n\n\n\n

$\\displaystyle\\int\\dfrac{\\left( \\sin ^{2} x\\right)^{2}-\\left(\\cos ^{2} x\\right)^{2}\\left\\{\\left(\\sin ^{2} x+\\cos ^{2} x\\right)^{2}-2 \\sin ^{2} x \\cdot \\cos ^{2} x \\right)}{1-2sin^2x.cos^2x}$<\/p>\n\n\n\n

$\\displaystyle\\int \\dfrac{\\left(\\sin ^{2} x-\\cos ^{2} x\\right)\\left(\\sin ^{2} x+\\cos ^{2} x\\right)\\left(1-2 \\sin ^{2} x-\\cos ^{2} x\\right) d x}{1-2sin^2x.cos^2x}$<\/p>\n\n\n\n

$=\\int\\left(\\sin ^{2} x-\\cos ^{2} x\\right) d x$<\/p>\n\n\n\n

$=-\\int\\left(\\cos ^{2} x-\\sin ^{2} x\\right) d x$<\/p>\n\n\n\n

$=-\\int \\cos 2 x d x$<\/p>\n\n\n\n

$=-\\frac{\\sin 2 x}{2}+c$<\/p>\n\n\n\n

Question 34<\/strong>
$\\displaystyle\\int \\dfrac{1+2 \\sin x}{\\cos ^{2} x} d x$
Sol :
$=\\int \\frac{1}{\\cos ^{2} x} d x+2 \\int \\frac{\\sin x}{\\cos x \\cdot \\cos x} d x$<\/p>\n\n\n\n

$=\\int \\sec ^{2} x d x+2 \\int \\sec x \\cdot \\tan x d x$<\/p>\n\n\n\n

=tanx+2secx+c<\/p>\n\n\n\n

Question 35<\/strong><\/p>\n\n\n\n

$\\int \\frac{5cos^{3} x+7 \\sin ^{2} x}{\\sin ^{2} x \\cdot \\cos ^{2} x} d x$<\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

=$5 \\int \\frac{\\cos ^{3} x}{\\sin^2 x.cos^2x}+7\\int \\frac{\\sin ^{3} x}{\\sin ^{2} x \\cdot \\cos ^{2} x}$<\/p>\n\n\n\n

$=5\\int \\frac{\\cos x}{\\sin x \\cdot \\sin x} d x+7\\int\\dfrac{sinx}{cosx.cosx}dx$<\/p>\n\n\n\n

=$5\\int \\frac{\\cos x}{\\sin x \\cdot \\sin x} d x+7 \\int \\frac{\\sin x}{\\cos x \\cdot \\cos x} d x$<\/p>\n\n\n\n

=$5 \\int \\cot x \\cdot \\operatorname{cosec} x d x+7 \\int \\tan x \\cdot \\sec x d x$<\/p>\n\n\n\n

=5(-cosecx)+7secx+c<\/p>\n\n\n\n

=7secx-5cosecx+c<\/p>\n\n\n\n

Question 36<\/strong><\/p>\n\n\n\n

$\\int \\frac{e^{x} \\sin x+\\cot x+x \\sin x}{\\sin x} d x$<\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

$\\int \\frac{e^{x} \\sin x}{\\sin x} d x+\\int \\frac{\\cot x}{\\sin x} d x+\\int \\frac{x \\sin x}{\\sin x} d x$<\/p>\n\n\n\n

$=\\int e^{x} d x+\\int \\cot x \\cdot \\operatorname{cosec} x+\\int x d x$<\/p>\n\n\n\n

$e^{x}-\\operatorname{cosec} x+\\frac{x^{2}}{2}+c$<\/p>\n\n\n\n

Question 37<\/strong><\/p>\n\n\n\n

$\\int \\sec (11+12 x) \\cdot \\tan (11+12 x) d x$
Sol :
$=\\frac{\\sec (11+12 x)}{12}+c$<\/p>\n\n\n\n

Question 38<\/strong><\/p>\n\n\n\n

$\\int(\\cos x+\\sin x)^{2} d x$<\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

=$\\int\\left(\\cos ^{2} x+\\sin^2 x+2 \\sin x \\cos x\\right) d x$<\/p>\n\n\n\n

$=\\int(1+\\sin 2 x) d x$<\/p>\n\n\n\n

$=\\int 1 d x+\\int \\sin 2 x d x$<\/p>\n\n\n\n

$=x-\\frac{\\cos 2 x}{2}+c$<\/p>\n\n\n\n

Question 39<\/strong><\/p>\n\n\n\n

If $\\frac{d y}{d x}=\\cos x+\\sec ^{2} x$ and where x=0 , y=0 what is y ?<\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

$f(x)=\\frac{d y}{d x}=\\cos x+\\sec ^{2} x$<\/p>\n\n\n\n

F(x) is antiderivative<\/p>\n\n\n\n

\u2234$F(x)=\\int f(x) d x=\\int \\frac{d y}{d x}=\\int \\cos x+\\sec ^{2} x d x$<\/p>\n\n\n\n

F(x)=sinx+tanx<\/p>\n\n\n\n

y=sinx+tanx<\/p>\n\n\n\n

Question 40<\/strong><\/p>\n\n\n\n

Find the antiderivative F of f<\/em> defined by f<\/em>(x)=4x2<\/sup>-6 where F(0)=3<\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

Antiderivative F(x)=$\\int f(x) d x=\\int 4 x^{3}-6 d x$<\/p>\n\n\n\n

$F(x)=4 \\int x^{3}-6 \\int d x$<\/p>\n\n\n\n

$=4 \\cdot \\frac{x^{4}}{4}-6 x+c$<\/p>\n\n\n\n

F(x)=x4<\/sup>-6x+c<\/p>\n\n\n\n

A.T.Q F(0)=04<\/sup>-6.0+c=3<\/p>\n\n\n\n

$F(x)=x^{4}-6 x+3$<\/p>\n\n\n\n

Question 41<\/strong><\/p>\n\n\n\n

If $f^{\\prime}(x)=4 x^{3}-\\dfrac{3}{x^4}$ , such that f<\/em>(2)=0 , then find f<\/em>(x)<\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

$f(x)=\\int f^{\\prime}(x) d x$<\/p>\n\n\n\n

$=\\int 4 x^{3}-\\frac{3}{x^{4}} d x$<\/p>\n\n\n\n

$f(x)=4 \\int x^{3} d x-3 \\int x^{-4} d x$<\/p>\n\n\n\n

$=4 \\frac{x^{4}}{4}-3 \\cdot \\frac{x^{-3}}{-3}+c$<\/p>\n\n\n\n

$f(x)=x^{4}+x^{-3}+c$$=x^{4}+\\frac{1}{x^{3}}+c$<\/p>\n\n\n\n

A.T.Q , f<\/em>(2)=$(2)^{4}+\\dfrac{1}{2^{3}}+c=0$<\/p>\n\n\n\n

$\\frac{129}{8}+c=0$<\/p>\n\n\n\n

$C=-\\frac{129}{8}$<\/p>\n\n\n\n

$f(x)=x^4+\\dfrac{1}{x^3}-\\dfrac{129}{8}$<\/p>\n\n\n\n

\n
KC Sinha Solutions<\/a><\/div>\n\n\n\n
KC Sinha Class 12 Solutions<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Question 1\u222b(x2-5x+7)dxSol := \u222bx2dx-5\u222bxdx+7\u222bdx [\u2235 \u222bxndx=$\\dfrac{x^{n+1}}{n+1}$] =$\\dfrac{x^3}{3}-\\dfrac{5x^2}{2}+7x+c$ Question 2 \u222b(ax3-bx2+cx+d)dxSol : =a\u222bx3dx+b\u222bx2dx+c\u222bxdx+d\u222bdx =$a\\dfrac{x^4}{4}+b\\dfrac{x^3}{3}+c\\dfrac{x^2}{2}+dx+c$ Question 3 $ \\displaystyle\\int \\left(x+\\dfrac{1}{x}+2\\right)$Sol := \u222bxdx+$\\int \\dfrac{1}{x}dx$+2\u222bdx =$\\dfrac{x^2}{2}$+log|x|+2x+c Question 4 \u222b(x1\/3+2×1\/2+x3\/2)dxSol :=\u222bx1\/3dx+2\u222bx1\/2dx+\u222bx3\/2dx =$\\dfrac{x^{\\frac{1}{3}+1}}{\\frac{1}{3}+1}+2 \\times \\dfrac{x^{\\frac{1}{2}+1}}{\\frac{1}{2}+1}+\\dfrac{x^{\\frac{3}{2}+1}}{\\frac{3}{2}+1}$+c =$\\dfrac{x^{4\/3}}{4\/3}+\\dfrac{2x^{3\/2}}{3\/2}+\\dfrac{x^{5\/2}}{5\/2}$+c =$\\dfrac{3}{4}x^{4\/3}+\\dfrac{2}{3}\\times 2 x^{3\/2}+\\dfrac{2}{5}x^{5\/2}$+c =$\\dfrac{3}{4}x^{4\/3}+\\dfrac{4}{3}x^{3\/2}+\\dfrac{2}{5}x^{5\/2}$+c Question 5 $\\displaystyle\\int \\dfrac{x^2-2x+3}{x^4}dx$ Sol : =$\\displaystyle\\int \\dfrac{x^2}{x^4}dx-2\\displaystyle\\int \\dfrac{x}{x^4}dx+3\\displaystyle\\int \\dfrac{1}{x^4}dx$ =$\\displaystyle\\int \\dfrac{1}{x^2}dx-2\\displaystyle\\int \\dfrac{1}{x^3}dx+3\\displaystyle\\int \\dfrac{1}{x^4}dx$ =$\\displaystyle\\int x^{-2}dx-2\\displaystyle\\int x^{-3}dx+3\\displaystyle\\int x^{-4}dx$ =$\\dfrac{x^{-2+1}}{-2+1}-2\\dfrac{x^{-3+1}}{-3+1}+3\\dfrac{x^{-4+1}}{-4+1}$+c =$\\dfrac{x^{-1}}{-1}-2\\dfrac{x^{-2}}{-2}+3\\dfrac{x^{-3}}{-3}$+c =$-\\dfrac{1}{x}+\\dfrac{1}{x^2}-\\dfrac{1}{x^3}$+c […]<\/p>\n","protected":false},"author":302,"featured_media":626470,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"newspack_featured_image_position":"","newspack_post_subtitle":"","newspack_article_summary_title":"Overview:","newspack_article_summary":"","newspack_hide_updated_date":false,"newspack_show_updated_date":false,"footnotes":""},"categories":[25],"tags":[],"boards":[],"class_list":["post-626464","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-class-12","entry"],"yoast_head":"\nKC Sinha: Exercise 19.1- Mathematics Solution Class 12 Chapter 19 Indefinite Integrals - IndCareer Schools<\/title>\n<meta name=\"description\" content=\"Question 1\u222b(x2-5x+7)dxSol := \u222bx2dx-5\u222bxdx+7\u222bdx =$dfrac{x^3}{3}-dfrac{5x^2}{2}+7x+c$ Question 2 \u222b(ax3-bx2+cx+d)dxSol : =a\u222bx3dx+b\u222bx2dx+c\u222bxdx+d\u222bdx\" \/>\n<meta name=\"robots\" content=\"index, follow, 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